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It is easy to see that if the function $f(z)$ has a primitive (i.e. antiderivative) then $$\int_{\Gamma} f(z)dz=0$$ for any closed curve $\Gamma$. Is the converse true, that is if the integral is $0$ for any closed curve $\Gamma$ then does it imply that $f(z)$ has a primitive ? Please provide a proof or a counterexample.

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If $f$ is continuous and its domain is connected, the answer is yes. This is known as Morera's Theorem.

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I think connectedness of domains can be disposed of as long as we require that every closed curve to be contained in one single component, but what about continuity? I can't think right now of a counterexample to Morera's Theorem: a function whose integral on any closed curve is zero in some domain but has no primitive...? – DonAntonio Sep 10 '12 at 12:13
@DonAntonio $f(z)=0$ if $z\ne0$, $f(0)=1$. I know it is a trivial counterexample, but it is still a counterexample. – Julián Aguirre Sep 10 '12 at 13:20

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